Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for the problems solved using alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

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Note that the width of objective \(i\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\):

## # A tibble: 4 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19255.      19233.      19224.      19012.      18703.
## 2     3      28356.      28479.      28149.      28047.      27084.
## 3     4      38029.      38259.      37875.        NaN         NaN 
## 4     5      47502.      47820.      47251.        NaN         NaN

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l         53640.   140
## 2 m         47876.   140
## 3 u         53710.   140
## 4 ul        49986.   140

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     2402.    40
##  2 m          2     1522.    40
##  3 u          2      598.    40
##  4 ul         2      792.    40
##  5 l          3    13578.    40
##  6 m          3     9871.    40
##  7 u          3     3471.    40
##  8 ul         3     5065.    40
##  9 l          4    75821.    30
## 10 m          4    78021.    30
## 11 u          4    78466.    30
## 12 ul         4    74500.    30
## 13 l          5   153191.    30
## 14 m          5   130213.    30
## 15 u          5   166755.    30
## 16 ul         5   150959.    30

Does \(m\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2    14414.    80
##  2 m          2    15134.    80
##  3 u          2    14192.    80
##  4 ul         2    13406.    80
##  5 l          3   148973.    40
##  6 m          3   131004.    40
##  7 u          3   157511.    40
##  8 ul         3   145011.    40
##  9 l          4    11207.    10
## 10 m          4     7464.    10
## 11 u          4     2700.    10
## 12 ul         4     4666.    10
## 13 l          5    28539.    10
## 14 m          5    17719.    10
## 15 u          5     5657.    10
## 16 ul         5     7844.    10

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

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